Question: Determine how many solutions exist for the system of equations. ${x-y = 9}$ ${-2x+2y = -18}$
Answer: Convert both equations to slope-intercept form: ${x-y = 9}$ $x{-x} - y = 9{-x}$ $-y = 9-x$ $y = -9+x$ ${y = x-9}$ ${-2x+2y = -18}$ $-2x{+2x} + 2y = -18{+2x}$ $2y = -18+2x$ $y = -9+x$ ${y = x-9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = x-9}$ ${y = x-9}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${x-y = 9}$ is also a solution of ${-2x+2y = -18}$, there are infinitely many solutions.